L(s) = 1 | + 3-s + 5-s + 2·7-s − 2·9-s − 11-s + 4·13-s + 15-s − 2·17-s + 2·21-s + 23-s − 4·25-s − 5·27-s − 7·31-s − 33-s + 2·35-s + 3·37-s + 4·39-s − 8·41-s + 6·43-s − 2·45-s − 8·47-s − 3·49-s − 2·51-s − 6·53-s − 55-s − 5·59-s + 12·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.208·23-s − 4/5·25-s − 0.962·27-s − 1.25·31-s − 0.174·33-s + 0.338·35-s + 0.493·37-s + 0.640·39-s − 1.24·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.134·55-s − 0.650·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.458816616\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.458816616\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.95575045127536336185363327794, −11.50606278511501469560040413868, −10.88023677523055899886301116177, −9.531323384279074079649336150802, −8.595888941067365837350463793393, −7.81893826163275369649161903506, −6.29323672924194667424631097798, −5.16796004190121525070885092557, −3.57632858228471856718550416679, −2.01168702564605014700712887669,
2.01168702564605014700712887669, 3.57632858228471856718550416679, 5.16796004190121525070885092557, 6.29323672924194667424631097798, 7.81893826163275369649161903506, 8.595888941067365837350463793393, 9.531323384279074079649336150802, 10.88023677523055899886301116177, 11.50606278511501469560040413868, 12.95575045127536336185363327794